Theorem lmodvs0 14335

Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
lmodvs0.f	|- F = (Scalar ` W )
lmodvs0.s	|- .x. = ( .s ` W )
lmodvs0.k	|- K = ( Base ` F )
lmodvs0.z	|- .0. = ( 0g ` W )

Assertion

Ref	Expression
lmodvs0	|- ( ( W e. LMod /\ X e. K ) -> ( X .x. .0. ) = .0. )

Proof of Theorem lmodvs0

Step	Hyp	Ref	Expression
1		lmodvs0.f	. . . . 5 |- F = (Scalar ` W )
2	1	lmodring 14308	. . . 4 |- ( W e. LMod -> F e. Ring )
3		lmodvs0.k	. . . . 5 |- K = ( Base ` F )
4		eqid 2231	. . . . 5 |- ( .r ` F ) = ( .r ` F )
5		eqid 2231	. . . . 5 |- ( 0g ` F ) = ( 0g ` F )
6	3, 4, 5	ringrz 14056	. . . 4 |- ( ( F e. Ring /\ X e. K ) -> ( X ( .r ` F ) ( 0g ` F ) ) = ( 0g ` F ) )
7	2, 6	sylan 283	. . 3 |- ( ( W e. LMod /\ X e. K ) -> ( X ( .r ` F ) ( 0g ` F ) ) = ( 0g ` F ) )
8	7	oveq1d 6032	. 2 |- ( ( W e. LMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( ( 0g ` F ) .x. .0. ) )
9		simpl 109	. . . 4 |- ( ( W e. LMod /\ X e. K ) -> W e. LMod )
10		simpr 110	. . . 4 |- ( ( W e. LMod /\ X e. K ) -> X e. K )
11	2	adantr 276	. . . . 5 |- ( ( W e. LMod /\ X e. K ) -> F e. Ring )
12	3, 5	ring0cl 14033	. . . . 5 |- ( F e. Ring -> ( 0g ` F ) e. K )
13	11, 12	syl 14	. . . 4 |- ( ( W e. LMod /\ X e. K ) -> ( 0g ` F ) e. K )
14		eqid 2231	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
15		lmodvs0.z	. . . . . 6 |- .0. = ( 0g ` W )
16	14, 15	lmod0vcl 14330	. . . . 5 |- ( W e. LMod -> .0. e. ( Base ` W ) )
17	16	adantr 276	. . . 4 |- ( ( W e. LMod /\ X e. K ) -> .0. e. ( Base ` W ) )
18		lmodvs0.s	. . . . 5 |- .x. = ( .s ` W )
19	14, 1, 18, 3, 4	lmodvsass 14326	. . . 4 |- ( ( W e. LMod /\ ( X e. K /\ ( 0g ` F ) e. K /\ .0. e. ( Base ` W ) ) ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. ( ( 0g ` F ) .x. .0. ) ) )
20	9, 10, 13, 17, 19	syl13anc 1275	. . 3 |- ( ( W e. LMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. ( ( 0g ` F ) .x. .0. ) ) )
21	14, 1, 18, 5, 15	lmod0vs 14334	. . . . 5 |- ( ( W e. LMod /\ .0. e. ( Base ` W ) ) -> ( ( 0g ` F ) .x. .0. ) = .0. )
22	17, 21	syldan 282	. . . 4 |- ( ( W e. LMod /\ X e. K ) -> ( ( 0g ` F ) .x. .0. ) = .0. )
23	22	oveq2d 6033	. . 3 |- ( ( W e. LMod /\ X e. K ) -> ( X .x. ( ( 0g ` F ) .x. .0. ) ) = ( X .x. .0. ) )
24	20, 23	eqtrd 2264	. 2 |- ( ( W e. LMod /\ X e. K ) -> ( ( X ( .r ` F ) ( 0g ` F ) ) .x. .0. ) = ( X .x. .0. ) )
25	8, 24, 22	3eqtr3d 2272	1 |- ( ( W e. LMod /\ X e. K ) -> ( X .x. .0. ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160  Scalarcsca 13162   .scvsca 13163   0gc0g 13338   Ringcrg 14008   LModclmod 14300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-sca 13175  df-vsca 13176  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-mgp 13933  df-ring 14010  df-lmod 14302

This theorem is referenced by:  lmodfopne  14339  lsssn0  14383